Average Error: 34.1 → 7.2
Time: 6.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.687169645099804665049460580076288185662 \cdot 10^{72}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.178383319658251659941996545038743748932 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 9.981010889976272049519072583984808542877 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.687169645099804665049460580076288185662 \cdot 10^{72}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.178383319658251659941996545038743748932 \cdot 10^{-298}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\

\mathbf{elif}\;b_2 \le 9.981010889976272049519072583984808542877 \cdot 10^{53}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r23056 = b_2;
        double r23057 = -r23056;
        double r23058 = r23056 * r23056;
        double r23059 = a;
        double r23060 = c;
        double r23061 = r23059 * r23060;
        double r23062 = r23058 - r23061;
        double r23063 = sqrt(r23062);
        double r23064 = r23057 - r23063;
        double r23065 = r23064 / r23059;
        return r23065;
}
↓
double f(double a, double b_2, double c) {
        double r23066 = b_2;
        double r23067 = -8.687169645099805e+72;
        bool r23068 = r23066 <= r23067;
        double r23069 = -0.5;
        double r23070 = c;
        double r23071 = r23070 / r23066;
        double r23072 = r23069 * r23071;
        double r23073 = 2.1783833196582517e-298;
        bool r23074 = r23066 <= r23073;
        double r23075 = 1.0;
        double r23076 = r23066 * r23066;
        double r23077 = a;
        double r23078 = r23077 * r23070;
        double r23079 = r23076 - r23078;
        double r23080 = sqrt(r23079);
        double r23081 = r23080 - r23066;
        double r23082 = r23075 / r23081;
        double r23083 = r23075 / r23070;
        double r23084 = r23082 / r23083;
        double r23085 = 9.981010889976272e+53;
        bool r23086 = r23066 <= r23085;
        double r23087 = -r23066;
        double r23088 = r23087 - r23080;
        double r23089 = r23077 / r23088;
        double r23090 = r23075 / r23089;
        double r23091 = 0.5;
        double r23092 = r23091 * r23071;
        double r23093 = 2.0;
        double r23094 = r23066 / r23077;
        double r23095 = r23093 * r23094;
        double r23096 = r23092 - r23095;
        double r23097 = r23086 ? r23090 : r23096;
        double r23098 = r23074 ? r23084 : r23097;
        double r23099 = r23068 ? r23072 : r23098;
        return r23099;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -8.687169645099805e+72

    1. Initial program 58.4

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 3.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -8.687169645099805e+72 < b_2 < 2.1783833196582517e-298

    1. Initial program 30.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--30.6

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified16.8

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified16.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

    8. Applied associate-/r*16.8

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}{a}}\]

    9. Simplified14.3

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}}{a}\]
    10. Using strategy rm

    11. Applied div-inv14.3

      \[\leadsto \frac{\frac{a}{\color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{c}}}}{a}\]

    12. Applied *-un-lft-identity14.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot a}}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{c}}}{a}\]

    13. Applied times-frac16.9

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{a}{\frac{1}{c}}}}{a}\]

    14. Applied associate-/l*16.4

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{a}{\frac{a}{\frac{1}{c}}}}}\]

    15. Simplified9.7

      \[\leadsto \frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{\frac{1}{c}}}\]

    if 2.1783833196582517e-298 < b_2 < 9.981010889976272e+53

    1. Initial program 9.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num9.8

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    if 9.981010889976272e+53 < b_2

    1. Initial program 37.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 5.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.687169645099804665049460580076288185662 \cdot 10^{72}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.178383319658251659941996545038743748932 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 9.981010889976272049519072583984808542877 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))